Bounded Languages Described by GF(2)-grammars
Vladislav Makarov
TL;DR
The paper develops an algebraic framework for GF(2)-grammars, a parity-based variant of context-free grammars, to study bounded languages. By translating languages into formal power series and embedding them in carefully constructed rings like $R_{a,b}$ and $R_{a,b,c}$, it derives strong necessary conditions and uses linear-algebraic arguments (via determinants and closed systems) to prove inherent ambiguity for several key languages, notably $\{a^n b^m c^n \mid n=m \text{ or } m=n\}$ and $\{a^n b^m c^\ell \mid n\neq m \text{ or } m\neq \ell\}$. Extending to general bounded languages through leftmost representations and GF(2)-transductions, the work shows that, in many cases, bounded-language describability reduces to one of a few canonical algebraic forms, providing a unified, algebraic route to longstanding ambiguity results and related closure properties. The paper also surveys related developments and outlines future directions in decidability, transductions, and connections to analytic methods.
Abstract
GF(2)-grammars are a recently introduced grammar family with some unusual algebraic properties. They are closely connected to unambiguous grammars. By using the method of formal power series, we establish strong conditions that are necessary for subsets of a^* b^* and a^* b^* c^* to be described by some GF(2)-grammar. By further applying the established results, we settle the long-standing open question of proving inherent ambiguity of the language {a^n b^m c^k | n != m or m != k}$, as well as give a new purely algebraic proof of the inherent ambiguity of the language {a^n b^m c^k}{n = m or m = k}.